nLab hypergroup

Idea

A hypergroup is a algebraic structure similar to a group, but where the composition operation does not just take two elements to a single product element in the group, but to a subset of elements of the group.

It is a hypermonoid with additional groupal structure and property.

Definition

A canonical hypergroup is a set, $H$, equipped with a commutative binary operation,

$$+ : H \times H \to \mathcal{P}(H)\backslash\{\emptyset\}$$

whose value is a non-empty subset of $H$, and a zero element $0 \in H$, such that

1. $+$ is associative (extended to allow addition of subsets of $H$);
2. $0 + x = {\{x\}} = x + 0, \forall x \in H$;
3. $\forall x \in H, \exists ! y \in H$ such that $0 \in x + y$ (we denote this $y$ as $-x$);
4. $\forall x, y, z \in H, x \in y + z$ implies $z \in x - y$ (where $x - y$ means $x + (-y)$ as usual).

$\mathcal{P}(H)$ is the notation for the power set of $H$. A related variant is the notion of n-valued group.

Examples

The additive structure underlying a hyperring is a canonical hypergroup. See there for more examples.

Related pages

• group

• hypermagma

• hypermonoid

• hyperring

• hypergraph

Remark

Beware that, while groups are equivalently pointed groupoids with a single object, hypergroups are not hypergroupoids with a single object. For this reason some sources refer to the latter as Duskin-Glenn hypergroupoids.

References

See also at hypermagma and multivalued group.

• J. Delsarte, Hypergroupes et opérateurs de permutation et de transmutation, La théorie des équations aux dérivées partielles. Nancy, 9-15 avril 1956, pp. 29–45 Colloq. Internat. CNRS, LXXI [International Colloquia of the CNRS] Centre National de la Recherche Scientifique, Paris, 1956 MR0116151, Zbl 0075.27503.

• J. Jantosciak, Transposition hypergroups: Noncommutative join spaces, J. Algebra 187 (1997) pp.97-119. doi:10.1006/jabr.1997.6789

• G. L. Litvinov, Hypergroups and hypergroup algebras, J. Math. Sci. 38 (1987) pp. 1734–1761, doi:10.1007/BF01088201 (Translated from Itogi Nauki i Tekhniki, Sovremennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 26, pp. 57–106, 1985), arXiv:1109.6596.

• F. Marty, Sur une généralization de la notion de groupe , IV Congrès des Mathématiciens Scandinaves, Stockholm 1934.

• P.-H. Zieschang, Hypergroups, ISBN 978-3-031-39488-1, XV+391 pages, Springer, Cham, 2023. doi:10.1007/978-3-031-39489-8

category: algebra